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Theorem foeq1 5045
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
foeq1 (𝐹 = 𝐺 → (𝐹:AontoB𝐺:AontoB))

Proof of Theorem foeq1
StepHypRef Expression
1 fneq1 4930 . . 3 (𝐹 = 𝐺 → (𝐹 Fn A𝐺 Fn A))
2 rneq 4504 . . . 4 (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
32eqeq1d 2045 . . 3 (𝐹 = 𝐺 → (ran 𝐹 = B ↔ ran 𝐺 = B))
41, 3anbi12d 442 . 2 (𝐹 = 𝐺 → ((𝐹 Fn A ran 𝐹 = B) ↔ (𝐺 Fn A ran 𝐺 = B)))
5 df-fo 4851 . 2 (𝐹:AontoB ↔ (𝐹 Fn A ran 𝐹 = B))
6 df-fo 4851 . 2 (𝐺:AontoB ↔ (𝐺 Fn A ran 𝐺 = B))
74, 5, 63bitr4g 212 1 (𝐹 = 𝐺 → (𝐹:AontoB𝐺:AontoB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  ran crn 4289   Fn wfn 4840  ontowfo 4843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-fo 4851
This theorem is referenced by:  f1oeq1  5060  foeq123d  5065  resdif  5091
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